# Why are continuous partial derivatives up to order two (rather than one) of nonlinear autonomous (2D) systems sufficient for linear approximation?

In Boyce and Diprima's ODE's and BVP's (10th edition page 522), it says that for the nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y) \qquad\qquad\qquad (10),$$

"The system (10) is locally linear in the neighborhood of a critical point $$(x_0,y_0)$$ whenever the functions $$F$$ and $$G$$ have continuous partial derivatives up to order two. To show this , we use Taylor expansions about the point $$(x_0,y_0)$$ to write $$F(x,y)$$ and $$G(x,y)$$ in the form $$F(x,y)=F(x_0,y_0)+F_x(x_0,y_0)(x-x_0)+F_y(x_0,y_0)(y-y_0)+\eta_1(x,y)\\ G(x,y)=G(x_0,y_0)+G_x(x_0,y_0)(x-x_0)+G_y(x_0,y_0)(y-y_0)+\eta_2(x,y)$$ where $$\eta_1(x,y)/[(x-x_0)^2+(y-y_0)^2]^{1/2}\to 0$$ as $$(x,y)\to(x_0,y_0)$$, and similarly for $$\eta_2$$."

Why does it say "whenever the functions $$F$$ and $$G$$ have continuous partial derivatives up to order two" instead of "whenever the functions $$F$$ and $$G$$ have continuous partial derivatives up to order one"? $$\eta_1(x,y)$$ and $$\eta_2(x,y)$$ contain the nonlinear parts and the second order partial derivatives, right? And, Taylor's Theorem says the function being k times differentiable is sufficient for an approximation by a kth degree Taylor polynomial.